Modular curve

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In number theory and algebraic geometry, a modular curve is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a finite index subgroup Γ of the modular group of integral 2×2 matrices. Modular curves are non-compact, but can be compactified by adding finitely many points "at infinity" called cusps. The points of a modular curve parametrize isomorphism classes of elliptic curves, together with some additional structure depending on the group Γ. This allows one to give a purely algebraic definition of modular curves, without reference to complex numbers, and, moreover, prove that modular curves are defined over the field Q of rational numbers or a cyclotomic field. The latter fact and its generalizations are of fundamental importance in number theory.

1 Analytic definition and examples
2 Relation with the Monster
3 Compactification
4 See also

[edit] Analytic definition and examples

The modular group acts on the upper half-plane by fractional linear transformations. The analytic definition of a modular curve involves a choice of a subgroup Γ satisfying an important technical condition. For any N ≥ 1, Γ(N) denotes the subgroup of the modular group consisting of matrices that are congruent to the identity matrix modulo N, it is called the principal congruence subgroup of level N . The technical condition is that Γ must contain the subgroup Γ(N) for some N, and the minimal such N is called the level of Γ. Then the modular curve X(Γ) is the noncompact Riemann surface

H/Γ.

The most common examples are the curves X(N) and X₀(N), associated with the groups Γ(N) and Γ₀(N), where the explicit classical model for X₀(N) is the classical modular curve; this is sometimes called the modular curve. The definition of Γ(N) can be restated as follows: it is the subgroup of the modular group which is the kernel of the reduction modulo N. Then Γ₀(N) is the larger subgroup consisting of matrices that are upper triangular modulo N. These curves have a direct interpretation as moduli spaces for elliptic curves, with 'markings'. Thus X(N) is the moduli space for elliptic curves with a given basis for the N-torsion, whereas X₀(N) is the moduli space for elliptic curves with a cyclic subgroup of order N. These curves have been studied in great detail, and in particular, it is known that X₀(N) can be defined over Q.

The equations defining modular curves are the best-known examples of modular equations. The best models can be very different from those taken directly from elliptic function theory. Hecke operators may be studied geometrically, as correspondences connecting pairs of modular curves.

[edit] Relation with the Monster

Modular curves of genus 0, which are quite rare, turned out to be of major importance in relation with the monstrous moonshine conjectures. In general a modular function field is a function field of a modular curve (or, occasionally, of some other moduli space that turns out to be an irreducible variety). Genus 0 means such a function field has a single transcendental function as generator: for example the j-function. The traditional name for such a generator, which is unique up to a Möbius transformation and can be appropriately normalized, is a Hauptmodul (main or principal modular function). First several coefficients of q-expansions of these functions were computed already in 19th century, but it came as a shock that the same large integers show up as dimensions of representations of the largest sporadic simple group Monster. The relation runs very deep and as demonstrated by Richard Borcherds, it also involves generalized Kac-Moody Lie algebras. Work in this area underlined the importance of modular functions that are meromorphic and can have poles at the cusps, as opposed to modular forms, that are holomorphic everywhere, including the cusps, and had been the main objects of study for the better part of the 20th century.

[edit] Compactification

The quotient

H/Γ

is not compact: for example, the path τ=yi, y>1 in the upper half-plane will project onto a path on the modular curve that does not have a limit. Compactification can be easily effected by considering the action of Γ on the extension H ∪ Q ∪ ∞i of the upper half-plane by "points at infinity". The equivalence classes of the points at infinity under the Γ action are called the cusps of Γ. They are finite in number, have neighborhoods analytically isomorphic to a complex disk, and thus the quotient

(H ∪ Q ∪ ∞i )/Γ

is a compact Riemann surface and complete algebraic curve, called a compactified modular curve.

Remark: quotients that are compact do occur for Fuchsian groups Γ other than subgroups of the modular group; a class of them constructed from quaternion algebras is also of interest in number theory.